The object of the present paper is to solve Fekete-Szegö problem and determine the sharp tipper bound to the second Hankel determinant for a certain class ℛλ(a, c, A, B) of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ℛλ(a, c, A, B) of ℛλ(a,c, A, B) and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

A certain general class 𝓢(a,c,A,B) of analytic functions involving a linear operator is introduced. The objective is to investigate various properties and characteristics of this class. Several applications of the results (obtained here) to a class of fractional calculus operators are also considered. The results contain some of the earlier work in univalent function theory.

Let = z ∈ ℂ; |z| < 1, T = z ∈ ℂ; |z|=1. Denote by S the class of functions f of the form f(z) = z + a₂z² + ... holomorphic and univalent in , and by S(M), M > 1, the subclass of functions f of the family S such that |f(z)| < M in . We introduce (and investigate the basic properties of) the class S(M,m;α), 0 < m ≤ M < ∞, 0 ≤ α ≤ 1, of bounded functions f of the family S for which there exists an open $arc{I}_{\alpha }=I_{\alpha }\left(f\right)\subset T$ of length 2πα such that ${\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le M$ for every $z₀\in I_{\alpha }$ and ${\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le m$ for every $z₀\in T{Ī}_{\alpha }$.

Let Mₚ denote the class of functions f of the form $f\left(z\right)=1/z^p+{\sum }_{k=0}^{\infty }aₖz^k$, p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n,p}\left(\alpha \right)=f:f\in Mₚ,Re-(z^{p+1}/p){\left(Dⁿf\right)}^{\text{'}}>\alpha$, α < 1, where $Dⁿf={\left(z^{n+p}f\left(z\right)\right)}^{\left(n\right)}/(z^{p}n!)$. Results on $L_{n,p}\left(\alpha \right)$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.

We geometrically define subclasses of starlike functions related to the class of uniformly starlike functions introduced by A. W. Goodman in 1991. We give an analytic characterization of these classes, some radius properties, and examples of functions in these classes. Our classes generalize the class of uniformly starlike functions, and many results of Goodman are special cases of our results.